# Construct two sequences so that $\lim_{n\rightarrow\infty } \frac{a_{n}}{b_{n}} \neq \lim_{n\rightarrow\infty } \frac{a}{b_{n}}$

Hey I am supposed to construct two sequences $$(a_n)_{1}^{\infty}$$ and $$(b_n)_{1}^{\infty}$$ , so that $$\lim_{n\rightarrow\infty } \frac{a_{n}}{b_{n}} \neq \lim_{n\rightarrow\infty } \frac{a}{b_{n}}$$, where $$\lim_{n\rightarrow\infty } a_n = a$$. I tried many different combinations with 1/n, but nothing worked so far. Thanks!

How about $$a_n=\frac{1}{n}$$ and $$b_n=\frac{1}{n^2}$$? Actually, $$a_n=b_n=\frac{1}{n}$$ works as well...

Take $$a_n = e^{-n}$$, so that $$a=0$$, and $$b_n = e^{-n}$$. The first limit is then $$1$$, whereas the second is $$0$$.

• This a nicer example than mine, since both limits are finite. – Mars Plastic Feb 9 at 18:36
• I do not see how it is nicer than yours, but thanks anyway. – Gibbs Feb 9 at 18:36

You can just take $$a_n=b_n=1/n$$. The left hand side is 1 and the right hand side is zero.